Bernoulli

  • Bernoulli
  • Volume 11, Number 6 (2005), 1059-1092.

On the convergence of the spectral empirical process of Wigner matrices

Z.D. Bai and J. Yao

Full-text: Open access

Abstract

It is well known that the spectral distribution Fn of a Wigner matrix converges to Wigner's semicircle law. We consider the empirical process indexed by a set of functions analytic on an open domain of the complex plane including the support of the semicircle law. Under fourth-moment conditions, we prove that this empirical process converges to a Gaussian process. Explicit formulae for the mean function and the covariance function of the limit process are provided.

Article information

Source
Bernoulli, Volume 11, Number 6 (2005), 1059-1092.

Dates
First available in Project Euclid: 16 January 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1137421640

Digital Object Identifier
doi:10.3150/bj/1137421640

Mathematical Reviews number (MathSciNet)
MR2189081

Zentralblatt MATH identifier
1101.60012

Keywords
central limit theorem linear spectral statistics random matrix spectral distribution Wigner matrices

Citation

Bai, Z.D.; Yao, J. On the convergence of the spectral empirical process of Wigner matrices. Bernoulli 11 (2005), no. 6, 1059--1092. doi:10.3150/bj/1137421640. https://projecteuclid.org/euclid.bj/1137421640


Export citation

References

  • [1] Bai, Z. (1993) Convergence rate of expected spectral distributions of large random matrices; Part I. Wigner matrices. Ann. Probab., 21, 625-648.
  • [2] Bai, Z. (1999) Methodologies in spectral analysis of large dimensional random matrices; A review. Statist. Sinica, 9, 611-677.
  • [3] Bai, Z. and Yin, Y. (1988) Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab., 16, 1729-1741.
  • [4] Bai, Z.D., Miao, B. and Tsay, J. (2002) Convergence rates of the spectral distributions of large Wigner matrices. Int. Math. J., 1(1), 65-90.
  • [5] Burkholder, D. (1973) Distribution function inequalities for martingales. Ann. Probab., 1, 19-42.
  • [6] Costin, O. and Lebowitz, J. (1995) Gaussian fluctuations in random matrices. Phys. Rev. Lett., 75, 69-72.
  • [7] Johansson, K. (1998) On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J., 91, 151-204.
  • [8] Khorunzhy, A.M., Khoruzhenko, B.A. and Pastur, L.A. (1996) Asymptotic properties of large random matrices with independent entries. J. Math. Phys., 37, 5033-5060.
  • [9] Rao, C.R. and Rao, M.B. (2001) Matrix Algebra and Its Applications to Statistics and Econometrics. River Edge, NJ: World Scientific.
  • [10] Sinai, Y. and Soshnikov, A. (1998) Central limit theorem for traces of large random symmetric matrices with independent matrix elements. Bol. Soc. Brasil. Mat. (N.S.), 29, 1-24.
  • [11] Titchmarsh, E. (1939) The Theory of Functions (2nd edn). London: Oxford University Press.
  • [12] Wigner, E. (1955) Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math., 62, 548-564.
  • [13] Wigner, E. (1958) On the distributions of the roots of certain symmetric matrices. Ann. Math., 67, 325-327.